53 research outputs found

### Deformations of the root systems and new solutions to generalised WDVV equations

A special class of solutions to the generalised WDVV equations related to a
finite set of covectors is investigated. Some geometric conditions on such a
set which guarantee that the corresponding function satisfies WDVV equations
are found (check-conditions). These conditions are satisfied for all root
systems and their special deformations discovered in the theory of the
Calogero-Moser systems by O.Chalykh, M.Feigin and the author. This leads to the
new solutions for the generalized WDVV equations.Comment: 8 page

### Locus configurations and $\vee$-systems

We present a new family of the locus configurations which is not related to
$\vee$-systems thus giving the answer to one of the questions raised in
\cite{V1} about the relation between the generalised quantum Calogero-Moser
systems and special solutions of the generalised WDVV equations. As a
by-product we have new examples of the hyperbolic equations satisfying the
Huygens' principle in the narrow Hadamard's sense. Another result is new
multiparameter families of $\vee$-systems which gives new solutions of the
generalised WDVV equation.Comment: 12 page

### Jack-Laurent symmetric functions for special values of parameters

Jack-Laurent symmetric functions for special values of parameter

### Yang-Baxter maps and integrable dynamics

The hierarchy of commuting maps related to a set-theoretical solution of the
quantum Yang-Baxter equation (Yang-Baxter map) is introduced. They can be
considered as dynamical analogues of the monodromy and/or transfer-matrices.
The general scheme of producing Yang-Baxter maps based on matrix factorisation
is discussed in the context of the integrability problem for the corresponding
dynamical systems. Some examples of birational Yang-Baxter maps coming from the
theory of the periodic dressing chain and matrix KdV equation are discussed.Comment: Revised version based on the talks at NEEDS conference (Cadiz, 10-15
June 2002) and SIDE-V conference (Giens, 21-26 June 2002

### A few things I learnt from Jurgen Moser

A few remarks on integrable dynamical systems inspired by discussions with
Jurgen Moser and by his work.Comment: An article for the special issue of "Regular and Chaotic Dynamics"
dedicated to 80-th anniversary of Jurgen Mose

### On non-QRT Mappings of the Plane

We construct 9-parameter and 13-parameter dynamical systems of the plane
which map bi-quadratic curves to other bi-quadratic curves and return to the
original curve after two iterations. These generalize the QRT maps which map
each such curve to itself. The new families of maps include those that were
found as reductions of integrable lattices

### Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems

A new construction using finite dimensional dual grassmannians is developed
to study rational and soliton solutions of the KP hierarchy. In the rational
case, properties of the tau function which are equivalent to bispectrality of
the associated wave function are identified. In particular, it is shown that
there exists a bound on the degree of all time variables in tau if and only if
the wave function is rank one and bispectral. The action of the bispectral
involution, beta, in the generic rational case is determined explicitly in
terms of dual grassmannian parameters. Using the correspondence between
rational solutions and particle systems, it is demonstrated that beta is a
linearizing map of the Calogero-Moser particle system and is essentially the
map sigma introduced by Airault, McKean and Moser in 1977.Comment: LaTeX, 24 page

### On Darboux-Treibich-Verdier potentials

It is shown that the four-parameter family of elliptic functions
$u_D(z)=m_0(m_0+1)\wp(z)+\sum_{i=1}^3 m_i(m_i+1)\wp(z-\omega_i)$ introduced
by Darboux and rediscovered a hundred years later by Treibich and Verdier, is
the most general meromorphic family containing infinitely many finite-gap
potentials.Comment: 8 page

### Discrete integrable systems and Poisson algebras from cluster maps

We consider nonlinear recurrences generated from cluster mutations applied to
quivers that have the property of being cluster mutation-periodic with period
1. Such quivers were completely classified by Fordy and Marsh, who
characterised them in terms of the skew-symmetric matrix that defines the
quiver. The associated nonlinear recurrences are equivalent to birational maps,
and we explain how these maps can be endowed with an invariant Poisson bracket
and/or presymplectic structure.
Upon applying the algebraic entropy test, we are led to a series of
conjectures which imply that the entropy of the cluster maps can be determined
from their tropical analogues, which leads to a sharp classification result.
Only four special families of these maps should have zero entropy. These
families are examined in detail, with many explicit examples given, and we show
how they lead to discrete dynamics that is integrable in the Liouville-Arnold
sense.Comment: 49 pages, 3 figures. Reduced to satisfy journal page restrictions.
Sections 2.4, 4.5, 6.3, 7 and 8 removed. All other results remain, with minor
editin

### Quantization of Solitons and the Restricted Sine-Gordon Model

We show how to compute form factors, matrix elements of local fields, in the
restricted sine-Gordon model, at the reflectionless points, by quantizing
solitons. We introduce (quantum) separated variables in which the Hamiltonians
are expressed in terms of (quantum) tau-functions. We explicitly describe the
soliton wave functions, and we explain how the restriction is related to an
unusual hermitian structure. We also present a semi-classical analysis which
enlightens the fact that the restricted sine-Gordon model corresponds to an
analytical continuation of the sine-Gordon model, intermediate between
sine-Gordon and KdV.Comment: 29 pages, Latex, minor updatin

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